Most of the useful fixes for drinking water need no tools and no landlord approval. Match the filter to the problem in your city's report, not to the longest list on the box.
Taste and chlorine
A basic carbon pitcher or faucet filter is enough. If your city uses chloramine, choose a filter that lists chloramine reduction, or expect it to work less well.
Lead
Pick a pitcher or faucet filter certified to NSF/ANSI Standard 53 for lead. Many popular pitchers are only certified for taste and chlorine under Standard 42. If you have young children or are pregnant and your building is older, test your tap.
PFAS
Countertop reverse osmosis units sit next to the sink and need no installation. Reverse osmosis is the most reliable home treatment for PFAS, and also handles lead, nitrate and arsenic. Some carbon pitchers carry NSF/ANSI 53 claims for PFOA and PFOS; check the certifier's listing.
Hard water
Renters can't usually install a softener. A shower filter will not stop scale. Small scale-inhibitor cartridges protect a kettle, coffee maker or dishwasher line; vinegar handles the rest.
Radon
Renters can test too: a $15-25 kit in a ground-floor or basement unit. Several states require landlords to disclose or address high radon; your state radon office can tell you the rules.